- Published 2017 in Discrete & Computational Geometry

A d-box is the cartesian product of d intervals of R and a d-box representation of a graph G is a representation of G as the intersection graph of a set of d-boxes in R. It was proved by Thomassen in 1986 that every planar graph has a 3-box representation. In this paper we prove that every graph embedded in a fixed orientable surface, without short non-contractible cycles, has a 5-box representation. This directly implies that there is a function f , such that in every graph of genus g, a set of at most f(g) vertices can be removed so that the resulting graph has a 5-box representation. We show that such a function f can be made linear in g. Finally, we prove that for any proper minor-closed class F , there is a constant c(F) such that every graph of F without cycles of length less than c(F) has a 3-box representation, which is best possible.

@article{Esperet2017BoxRO,
title={Box Representations of Embedded Graphs},
author={Louis Esperet},
journal={Discrete & Computational Geometry},
year={2017},
volume={57},
pages={590-606}
}